Great question! The issue here is the difference between how English and Hindi express "standing": In English, it's usually seen as an action (like running, walking, etc.), and hence the idea (of "standing") is conveyed using a verb, whereas in Hindi, it's seen as a state (like happy, tall, etc.), and hence requires an adjective. This is one of those cases where the same concept is expressed in different languages by different parts of speech.

I am happy. (adj) - मैं ख़ुश हूँ। (adj)

I am running. (vrb) - मैं दौड़ रहा हूँ। (vrb)

I am standing. (vrb) - मैं खड़ा हूँ। (adj)

On a side note, "(हम) खड़े रहे हैं" by itself is a grammatically valid construction and it means "(we) have been standing", although it's highly unlikely that you would ever hear such an expression. The present perfect continuous tense isn't generally used in Hindi and it's almost always the present continuous tense that's used instead. So,

We have been standing since morning. - हम सुबह से खड़े हैं। (Lit: We are standing since morning.)

Forget about the -ing for a minute. Take "sit" as the example, first.

मैं बैठा हूँ

हम बैठे हैं

These mean "I am SEATED / We are SEATED" बैठा, बैठे etc are the "perfect" aspect of the verb -- They indicate COMPLETED action. In Hindi, "sit" is either completed or not completed; on or off. Same with "stand." So,

मैं खड़ा हूँ

हम खड़े हैं

I am STOOD / We are STOOD

it's DL's problem , im a native speaker and i know हम आज सड़क पर खड़े रहे हैं is correct. so maybe the contributors forgot to add this meaning! :D